quasi
Posts:
9,092
Registered:
7/15/05
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Re: A good probability puzzle but what is the right wording?
Posted:
Feb 4, 2013 8:10 AM
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Paul wrote:
>
>The following puzzle is copied and pasted from the internet.
>
> Alice secretly picks two different real numbers by an unknown
> process and puts them in two (abstract) envelopes. Bob
> chooses one of the two envelopes randomly (with a fair coin
> toss), and shows you the number in that envelope. You must
> now guess whether the number in the other, closed envelope
> is larger or smaller than the one you?ve seen. Is there a
> strategy which gives you a better than 50% chance of guessing
> correctly, no matter what procedure Alice used to pick her
> numbers?
Yes.
Let R denote the set of real numbers and let (0,1) denote
the open interval from 0 to 1.
Let f : R -> (0,1) be a strictly decreasing function.
Use the following strategy:
If the initially exposed value is t, "switch" with probability
f(t) and "stay" with probability 1 - f(t).
Suppose Alice chooses the pair x,y with x < y (by whatever
process, it doesn't matter). After Alice choose that pair,
then, by following the strategy I specified above, the
probability of guessing the highest card is exactly
(1/2)*f(x) + (1/2)*(1 - f(y))
which simplifies to
1/2 + f(x) - f(y)
and that exceeds 1/2 since f is strictly decreasing.
Of course, it's not the case that probability of guessing
correctly is more than c for any fixed c > 1/2, but the
problem didn't require that.
quasi
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